Question

5. Sum of the roots of a quadratic
equation is double their product. Find k if equation
is x2 - 4x + k + 3 = 0





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Answer 1

Let,

the roots of the quadratic equation be $\alpha$and$\beta$

sum of roots =$\alpha+\beta$

product of roots =$\alpha\times\beta$

Given that,

sum of roots = double the product

i.e $\alpha+\beta=2\alpha\beta$...(l)

Given : ${x}^{2}-4kx+k+3 =0$....(ll)

General form of quadratic equation is,

${x}^{2}-(sum\:of\:root)x+product\:of\:roots=0\\\\→{x}^{2}-(\alpha+\beta)x+\alpha\beta=0\\\\→{x}^{2}-(2\alpha\beta)x+\alpha\beta=0$....(lll)

from equation (ll) and (lll)

$4k=2\alpha\beta \:and\: k+3=\alpha\beta$

solving these equations we have,

$4k=2(k+3)\\→2k=k+3\\→k=3$